The grounding should be fine...
What you're doing here is running multiple circuits (the feeder out to the subpanel and the branch circuits back) through a single raceway with a single equipment grounding conductor, as implicitly provided for in NEC 250.122(C):
(C) Multiple Circuits. Where a single equipment grounding conductor is run with multiple circuits in the same
raceway, cable, or cable tray, it shall be sized for the largest
overcurrent device protecting conductors in the raceway,
cable, or cable tray. Equipment grounding conductors installed in cable trays shall meet the minimum requirements
of 392.10(B)(1)(c).
This, plus the fact that a single EGC can branch into multiple EGCs downstream of a panel when a multi-circuit raceway splits into individual circuits, means that you are fine on the grounding standpoint.
However, that poor, sweltering feeder cable isn't!
That feeder cable, on the other hand, is very not-fine. Putting 18 current-carrying wires (8 branch circuit hots and neutrals each, as well as the 2 feeder hots) into a single raceway forces a whopping 50% derate on the conductors (as per Table 310.15(B)(3)(a)), and 30" is a bit too long for the short-raceway provisions in 310.15(B)(3)(a) point 2, which are limited to raceways 24" or shorter. This means that your 2/2/2/4 SE cable is limited to 50A(!) and your 12AWG branch circuit wires can only handle 15A.
Fixing this would require re-running the middle panel's feeder and returning branch circuits using properly sized wires -- the branch-circuit wires would need to be bumped up to 10AWG, and the feeder hot wires would need to be bumped up to 1/0AWG (although a smaller neutral could be acceptable here, depending on the amount of unbalanced load anticipated in the subpanel). With your 1.5" EMT, the 10AWG THHN branch circuit wires (at 13.61mm2 of fill each), and compact stranded XHHW-2 singles for the feeder hot conductors, this leaves you with 422.96mm2 of fill used out of 526mm2 usable, before accounting for the neutral -- a full-size 1/0AWG neutral takes up another 102.6mm2 of fill, pushing you to the very limit of what that conduit can handle.
As a result, I'd recommend replacing it with 2" EMT to provide room for growth, or running two more EMTs (1/2" will do, with 72mm2 used vs 78mm2 usable) to allow the use of 12AWG wire for the branch circuits coming back via the new conduits. (Adding the extra conduits doesn't change the grounding logic from above at all, by the way.)
Best Answer
Frame challenge:
You need a generator interlock. Full stop.
You need a mechanical device that positively ensures the generator cannot backfeed the power grid.
You cannot replace this with a checklist, or with "being super smart". The rest of us need to do that, and you are not smarter than the rest of us - especially not during a power failure when you're running around trying to get the lights back on, in the dark, in the rain.
If you doubt that, consider the difficulty you're having solving this problem, in good conditions. You are not Superman and you CAN make mistakes. The interlock's job is to ensure those mistakes don't kill someone. And it is mandatory.
So the idea of backfeeding your pool subpanel is Right Out, because it makes the proper interlock impossible. You are either going to have to
a) lay a new cable, or b) "permanently" re-task the pool supply cable to be the generator supply cable, resulting in the pool ONLY being powerable off the generator.
Yes, I'm aware that's exactly what you're trying to avoid. The rest of us don't get to do that, neither do you.
I say "permanently" because there is nothing wrong with doing permanent and proper wiring seasonally, as long as it is proper and to Code.
Now, if you get us some information about your main panel, we can guide you to some good choices in generator interlock. (We're not normally a shopping site, but we can give some guidance).
Why?
Because power flows both ways, as you well know from your efforts to backfeed the pool sub. It also backfeeds out onto the neighborhood grid. It also backfeeds through transformers. When you energize the 120/240V side of a 120/240--9600V transformer, the high side of the transformer energizes at -- anyone, anyone, Bueller? Correct, 9600 volts.
And that's how you kill linemen with a Harbor Freight generator.